Difference between revisions of "Single Transformations of Functions"

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==Introduction==
 
==Introduction==
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<div class="floatright">
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<gallery perrow="2" widths="300px" heights="200px">
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File:Vertical shift.png|Vertical shift: <math> f(x) = x^2 </math> (red) and <math> g(x) = x^2 + 4 </math> (blue)
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File:Horizontal shift.png|Horizontal shift: <math> f(x) = sin(x) </math> (red) and <math> g(x) = sin(x + \frac{\pi}{2}) </math> (blue)
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File:Vertical reflection.png|Vertical reflection: <math> f(x) = \sqrt{x} </math> (red) and <math> g(x) = -\sqrt{x} </math> (blue)
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File:Horizontal reflection.png|Horizontal reflection: <math> f(x) = e^x </math> (red) and <math> g(x) = e^{-x} </math> (blue)
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File:Vertical stretch compression.png|Vertical stretch/compression: <math> f(x) = x^2 </math> (red), <math> g(x) = \frac{1}{2}x^2 </math> (blue, vert. compression), <math> h(x) = 2x^2 </math> (green, vert. stretch), and <math> j(x) = -2x^2 </math> (black, vert. stretch and vert. reflection)
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File:Horizontal stretch compression.png|Horizontal stretch/compression: <math> f(x) = \sqrt{x} </math> (red), <math> g(x) = \sqrt{2x} </math> (blue, horiz. compression), <math> h(x) = \sqrt{\frac{1}{2}x} </math> (green, horiz. stretch), and <math> j(x) = \sqrt{-2x} </math> (black, horiz. compression and horiz. reflection)
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</gallery>
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</div>
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===Translations===
 
One kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function <math> g(x) = f(x) + k </math>, the function <math> f(x) </math> is shifted vertically k units. For example, <math> g(x) = x^2 + 4 </math> is the function <math> f(x) = x^2 </math> shifted up by 4 units. <math> g(x) = sin(x) - 7.7 </math> is the function <math> f(x) = sin(x) </math> shifted down by 7.7 units.
 
One kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function <math> g(x) = f(x) + k </math>, the function <math> f(x) </math> is shifted vertically k units. For example, <math> g(x) = x^2 + 4 </math> is the function <math> f(x) = x^2 </math> shifted up by 4 units. <math> g(x) = sin(x) - 7.7 </math> is the function <math> f(x) = sin(x) </math> shifted down by 7.7 units.
  
 
Given a function f, a new function <math> g(x) = f(x - h) </math>, where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will shift h units to the right. If h is negative, the graph will shift h units to the left. For example, <math> g(x) = (x - 3)^2 </math> is the graph of <math> f(x) = x^2 </math> shifted 3 units to the right. <math> g(x) = sin(x + \frac{\pi}{2}) </math> is the function <math> f(x) = sin(x) </math> shifted <math> \frac{\pi}{2} </math> units to the left.
 
Given a function f, a new function <math> g(x) = f(x - h) </math>, where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will shift h units to the right. If h is negative, the graph will shift h units to the left. For example, <math> g(x) = (x - 3)^2 </math> is the graph of <math> f(x) = x^2 </math> shifted 3 units to the right. <math> g(x) = sin(x + \frac{\pi}{2}) </math> is the function <math> f(x) = sin(x) </math> shifted <math> \frac{\pi}{2} </math> units to the left.
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===Reflections===
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Given a function <math> f(x) </math>, a new function <math> g(x) = -f(x) </math> is a vertical reflection of the function <math> f(x) </math>, sometimes called a reflection about (or over, or through) the x-axis. For example, <math> g(x) = -\sqrt{x} </math> is a vertical reflection of the function <math> f(x) = \sqrt{x}</math>.
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Given a function <math> f(x) </math>, a new function <math> g(x) = f(-x) </math> is a horizontal reflection of the function <math> f(x) </math>, sometimes called a reflection about the y-axis. For example, <math> g(x) = e^{-x} </math> is a horizontal reflection of the function <math> f(x) = e^x </math>.
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===Even and Odd Functions===
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A function f is even if for all values of x, <math> f(x) = f(-x) </math>; that is, a function <math> f(x) </math> is even if its horizontal reflection <math> f(-x) </math> is identical to itself. For example, <math> f(x) = x^2 </math> is an even function since <math> f(-x) = (-x)^2 = (-1)^2(x)^2 = x^2 = f(x)</math>.
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A function f is odd if for all values of x, <math> f(x) = -f(-x) </math>; that is, a function <math> f(x) </math> is odd if a horizontal reflection and vertical reflection of the function results in the same function. For example, <math> f(x) = x^3 </math> is an odd function since <math> -f(-x) = -(-x)^3 = (-1)(-1)^3(x)^3 = x^3 = f(x)</math>.
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If a function satisfies neither of these conditions, then it is neither even nor odd. For example, <math> f(x) = x^2 + x </math> is neither even nor odd because <math> f(-x) = (-x)^2 + (-x) = x^2 - x </math>, which is not equal to <math> f(x) </math>, and <math> -f(-x) = -(-x)^2 - (-x) = -x^2 + x </math>, which is also not equal to <math> f(x) </math>.
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===Compressions and Stretches===
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Given a function <math> f(x) </math>, a new function <math> g(x) = af(x) </math>, where <math> a </math> is a constant, is a vertical stretch or vertical compression of the function <math> f(x) </math>.
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* If <math> a > 1 </math>, then the graph will be stretched.
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* If <math> 0 < a < 1 </math>, then the graph will be compressed.
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* If a < 0, then there will be a vertical stretch or compression of a factor of <math> |a| </math>, along with a vertical reflection.
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Given a function <math> f(x) </math>, a new function <math> g(x) = f(bx) </math>, where <math> b </math> is a constant, is a horizontal stretch or horizontal compression of the function <math> f(x) </math>.
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* If <math> b > 1 </math>, then the graph will be horizontally compressed by a factor of <math> \frac{1}{b} </math>.
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* If <math> 0 < b < 1 </math>, then the graph will be horizontally stretched by a factor of <math> \frac{1}{b} </math>.
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* If b < 0, then there will be a horizontal stretch or compression of a factor of <math> \Big|\frac{1}{b} \Big| </math>, along with a vertical reflection.
  
 
==Resources==
 
==Resources==
 
* [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-transformations-of-functions/ Intro to Transformations of Functions], Lumen Learning
 
* [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-transformations-of-functions/ Intro to Transformations of Functions], Lumen Learning
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== Licensing ==
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Content obtained and/or adapted from:
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* [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-transformations-of-functions/ Transformations of Functions, Lumen Learning College Algebra] under a CC BY-SA license

Latest revision as of 12:39, 15 January 2022

Introduction


Translations

One kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function , the function is shifted vertically k units. For example, is the function shifted up by 4 units. is the function shifted down by 7.7 units.

Given a function f, a new function , where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will shift h units to the right. If h is negative, the graph will shift h units to the left. For example, is the graph of shifted 3 units to the right. is the function shifted units to the left.

Reflections

Given a function , a new function is a vertical reflection of the function , sometimes called a reflection about (or over, or through) the x-axis. For example, is a vertical reflection of the function .

Given a function , a new function is a horizontal reflection of the function , sometimes called a reflection about the y-axis. For example, is a horizontal reflection of the function .

Even and Odd Functions

A function f is even if for all values of x, ; that is, a function is even if its horizontal reflection is identical to itself. For example, is an even function since .

A function f is odd if for all values of x, ; that is, a function is odd if a horizontal reflection and vertical reflection of the function results in the same function. For example, is an odd function since .

If a function satisfies neither of these conditions, then it is neither even nor odd. For example, is neither even nor odd because , which is not equal to , and , which is also not equal to .

Compressions and Stretches

Given a function , a new function , where is a constant, is a vertical stretch or vertical compression of the function .

  • If , then the graph will be stretched.
  • If , then the graph will be compressed.
  • If a < 0, then there will be a vertical stretch or compression of a factor of , along with a vertical reflection.

Given a function , a new function , where is a constant, is a horizontal stretch or horizontal compression of the function .

  • If , then the graph will be horizontally compressed by a factor of .
  • If , then the graph will be horizontally stretched by a factor of .
  • If b < 0, then there will be a horizontal stretch or compression of a factor of , along with a vertical reflection.

Resources

Licensing

Content obtained and/or adapted from: